In this regime, the WKB (adiabatic) approximation is not valid anymore and the axion energy density is diluted as \(\propto a^{-6}\)  due to the expansion of the Universe (as illustrated in Fig. REF ). This can be shown by noting that \(\dot{\theta }_a f_a^2\)  is the Noether charge density associated to the PQ symmetry [1]} and therefore it decays as \(\dot{\theta }_a \propto a^{-3}\)  as long as the axion mass can be
neglected.Alternatively, the dilution of the axion velocity can be obtained
from the axion EOM in the massless limit:
\(\ddot{\theta }_a+3 H \dot{\theta }_a=0\,\Rightarrow \dot{\theta }_a a^3={\rm const}\) . In consequence, a comoving PQ charge \(q_{\rm kin}\)  can be defined which is a conserved quantity,
\(q_{\rm kin}=\dot{\theta }_a a^3={\rm const} \, .\) 
